I've got same problem with mine, HP says it's hardware related and will charge $398 if it is out of warranty to fix it. I have an HP ze4400, but all of our internals are the same, just different badging. I have narrowed it down to the MoBo, which is seeming to be the hardest thing to find a replacement for. Feel free to hit me up if you got any other ?'s

One thing comes to mind and it sounds like most of the same thing I ran into.
1 if the battery is very dead you cannot run the laptop and charge at the same time the smaller power supplies will not handle the out draw.
2 since you say it will also not run on just the power adaptor with out the battery but will work somewhat with both in this is my thought.
I would suspect the power supply not having enough juice (amps)
and your battery has run down. This I would check before I go any further.
Find a good known good power supply (take out) your battery and plug it in. sometime the power supply will show the right voltage amps are not enough to power the unit. Do the same with the other laptop using your power supply if available.
If the other laptop still works and yours is not then you will need to go deeper I this case as well I would go to the power circuit first.
Alex